1. Probabilistic and Statistical Techniques 1 Lecture 3 Eng. Ismail Zakaria El Daour 2010

2. 2 - Overview - Frequency Distributions - Histograms - Statistical Graphics Probabilistic and Statistical Techniques Summarizing and Graphing Data

3. 3 1. Center: A representative or average value that indicates where the middle of the data set is located. 2. Variation: A measure of the amount that the values vary among themselves. 3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed). 4. Outliers: Sample values that lie very far away from the vast majority of other sample values. Overview Important Characteristics of Data Probabilistic and Statistical Techniques

4. 4 Key Concept When working with large data sets, it is often helpful to organize and summarize data by constructing a table called a frequency distribution, defined later. Because computer software and calculators can generate frequency distributions, the details of constructing them are not as important as what they tell us about data sets. Probabilistic and Statistical Techniques Frequency Distributions

5. 5 Frequency Distribution (or Frequency Table) lists data values (either individually or by groups of intervals), along with their corresponding frequencies or counts Definition Probabilistic and Statistical Techniques

6. 6 Frequency Distribution Ages of Best Actresses Original Data Probabilistic and Statistical Techniques

7. 7 Frequency Distribution Ages of Best Actresses Frequency Distribution Probabilistic and Statistical Techniques

8. 8 are the smallest numbers that can actually belong to different classes Lower Class Limits Lower Class Limits Probabilistic and Statistical Techniques

9. 9 Upper Class Limits are the largest numbers that can actually belong to different classes Upper Class Limits Probabilistic and Statistical Techniques

10. 10 are the numbers used to separate classes, but without the gaps created by class limits Class Boundaries Editor: Substitute Table 2-2 Class Boundaries 20.5 30.5 40.5 50.5 60.5 70.5 80.5 Probabilistic and Statistical Techniques

11. 11 Class Midpoints can be found by adding the lower class limit to the upper class limit and dividing the sum by two Class Midpoints 25.5 35.5 45.5 55.5 65.5 75.5 Probabilistic and Statistical Techniques

12. 12 Class Width is the difference between two consecutive lower class limits or two consecutive class boundaries Editor: Substitute Table 2-2 Class Width 10 Probabilistic and Statistical Techniques

13. 13 1. Large data sets can be summarized. 2. We can gain some insight into the nature of data. 3. We have a basis for constructing important graphs such as Histogram. Reasons for Constructing Frequency Distributions Probabilistic and Statistical Techniques

14. 14 3. Starting point: Begin by choosing a lower limit of the first class. 4.Using the lower limit of the first class and class width, proceed to list the lower class limits. 5. List the lower class limits in a vertical column and proceed to enter the upper class limits. 6. Go through the data set putting each data value in its class interval Constructing A Frequency Distribution 1. Decide on the number of classes (should be between 5 and 20). 2. Calculate class width class width  (maximum value) – (minimum value) number of classes Probabilistic and Statistical Techniques

15. 15 Constructing A Frequency Distribution 1. Decide on the number of classes (should be between 5 and 20). A useful recipe to determine the number of classes (K) is the “2 to the k rule”. This guide suggests you select the smallest number k for the number of classes such that 2 k is greater than the number of observations n. Probabilistic and Statistical Techniques

16. 16 Relative Frequency Distribution Relative Frequency = class frequency sum of all frequencies includes the same class limits as a frequency distribution, but relative frequencies are used instead of actual frequencies Probabilistic and Statistical Techniques

17. 17 Relative Frequency Distribution Total Frequency = 76 Probabilistic and Statistical Techniques

18. 18 Relative Frequency Distribution 28/76 = 37% 30/76 = 39% etc. Probabilistic and Statistical Techniques

19. 19 Cumulative Frequency Distribution Probabilistic and Statistical Techniques

20. 20 Cumulative Frequencies Cumulative Frequency Distribution Probabilistic and Statistical Techniques

21. 21 Frequency Tables Probabilistic and Statistical Techniques

22. 22 Critical Thinking Interpreting Frequency Distributions In later chapters, there will be frequent reference to data with a normal distribution. One key characteristic of a normal distribution is that it has a “bell” shape. The frequencies start low, then increase to some maximum frequency, then decrease to a low frequency. The distribution should be approximately symmetric. Probabilistic and Statistical Techniques

23. 23 Key Concept A histogram is an important type of graph that portrays the nature of the distribution. Probabilistic and Statistical Techniques Histograms

24. 24 Histogram A bar graph in which the horizontal scale represents the classes of data values and the vertical scale represents the frequencies Probabilistic and Statistical Techniques

25. 25 Relative Frequency Histogram Has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies instead of actual frequencies Probabilistic and Statistical Techniques

26. 26 Critical Thinking One key characteristic of a normal distribution is that it has a “bell” shape. The histogram below illustrates this. Probabilistic and Statistical Techniques

27. Application Examples : 27

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31. 31 frequency diagram relative frequency diagram

32. Examples of histograms Probabilistic and Statistical Techniques

33. Examples of histograms Probabilistic and Statistical Techniques

34. Examples of histograms Probabilistic and Statistical Techniques

35. Examples of histograms Probabilistic and Statistical Techniques

36. Examples of histograms Probabilistic and Statistical Techniques

37. Probability distribution Probabilistic and Statistical Techniques

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